The invention relates generally to methods and apparatus for computing a computationally intensive algorithm, and more specifically to a method and apparatus for computing log(x), or equivalently, −log(x), in a manner that is particularly useful for computed tomographic image processing and other applications.
In at least one known computed tomography (CT) imaging system configuration, an x-ray source projects a fan-shaped beam which is collimated to lie within an X-Y plane of a Cartesian coordinate system and generally referred to as the “imaging plane.” The x-ray beam passes through the object being imaged, such as a patient. The beam, after being attenuated by the object, impinges upon an array of radiation detectors. The intensity of the attenuated beam radiation received at the detector array is dependent upon the attenuation of the x-ray beam by the object. Each detector element of the array produces a separate electrical signal that is a measurement of the beam attenuation at the detector location. The attenuation measurements from all the detectors are acquired separately to produce a transmission profile.
In known third generation CT systems, the x-ray source and the detector array are rotated with a gantry within the imaging plane and around the object to be imaged so that the angle at which the x-ray beam intersects the object constantly changes. A group of x-ray attenuation measurements, i.e., projection data, from the detector array at one gantry angle is referred to as a “view.” A “scan” of the object comprises a set of views made at a different gantry angles, or view angles, during one revolution of the x-ray source and detector. In an axial scan, the projection data is processed to construct an image that corresponds to a two dimensional slice taken through the object. One method for reconstructing an image from a set of projection data is referred to in the art as the filtered back projection technique. This process converts the attenuation measurements from a scan into intergers called “CT numbers” or “Hounsfield units,” which are used to control the brightness of a corresponding pixel on a cathode ray tube display.
The negative natural logarithm function −log(x) is an important yet computationally intensive algorithm in computed tomographic (CT) image processing. In known systems, a 5th order polynomial is used to approximate the function. However, this polynomial still consumes more than 20% of the total image processing time and generates a relatively large approximation error and error standard deviation.
A positive floating point variable x can be represented by an expression written as:x=m×2e  (1)                where m (1≦m<2) is a mantissa and e is a binary exponent.        
Using equation (1), −log(x) can be written as:y=−log(x)=−log(m)−e×log(2)  (2)
The following equation uses a finite order polynomial to approximate log(m) in a region 1≦m<2. Generally speaking, the higher the order of the polynomial, the better the approximation will be, but the computational load is in proportion to the order of the polynomial. For example, a 5th order polynomial presently used is written as:log(m)≈(a0+a1m+a2m2+a3m3+a4m4+a5m5)  (3a)or as:y=−log(x)≈−(a0+m(a1+m(a2+m(a3+m(a4+a5m))))+e×a6)  (3b)
In equation (3b), a0˜a6 are precalculated constants. To compute −log(x), six additions and six multiplications are required, plus mantissa and exponent extractions.
To process images more efficiently and accurately, it would be desirable to provide methods and apparatus to reduce the complexity of the approximation used to calculate −log(x) while achieving numerical accuracy consistent with IEEE (Institute of Electrical and Electronic Engineers) floating-point precision.